Finite Group Databasebeta

← #13144508

Subgroups — #13144508

Order 1728.

836 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722664C211yesmaximal, characteristic
2H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
3#11721859C212yesmaximal, complemented
4#11720579C211yesmaximal, characteristic, complemented
5#11720522C212yesmaximal, complemented
6C3×#5061—34nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—22no—
10#88744—22no—
11D4×#272—22no—
12#88740C2212yes—
13C22⋊C4×H(3,𝔽3)—22no—
14C42×H(3,𝔽3)C2211yescharacteristic, complemented
15#88550—22no—
16#88549—22no—
17#88538C2212yescomplemented
18C3×#1004—34no—
19C32×#124D314yescomplemented
20C3×#942—38no—
21C3×#833—34no—
22C3×#805—38no—
23C23×H(3,𝔽3)D412yes—
24C22×#272—22no—
25Q8×H(3,𝔽3)—22no—
26D4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30#6235—24no—
31#6235—22no—
32C2×#1084—22no—
33C4⋊#272D412yes—
34C4×#272—22no—
35#6229D412yes—
36#6180—22no—
37C3×#455—91no—
38#5061—34no—
39C2×C32×Q8C2×D314yescomplemented
40C2×C32×D4C2×D314yes—
41C2×C3×#67—68no—
42C3×#237—68no—
43C3×D3×D4—68no—
44C3×#235—38no—
45C32×C22⋊C4—28no—
46C32×C42C2×D314yescomplemented
47C3×#216—68no—
48C3×#215—68no—
49C3×#208—38no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—42no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C2×#272—22no—
54C2×#272—82no—
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)—41no—
57C4×H(3,𝔽3)C2×D412yes—
58#1084—22no—
59C3×H(5,𝔽2)—91no—
60#1004—34no—
61C3×#137—92no—
62C3×#124—34no—
63C3×#124C3⋊D311yescharacteristic
64#942—38no—
65C3×#104—92no—
66C3×#100—91no—
67#833—34no—
68#805—38no—
69C23×C32#6718yes—
70C22×C3×D3—68no—
71C32×Q8—28no—
72C32×D4—28no—
73C2×C32×C4—24no—
74C2×C32×C4—24no—
75C2×C32×C4C22×D314yes—
76C3×#67—616no—
77C3×#67—68no—
78C2×C3×Dic3—68no—
79C3×D12—38no—
80C3×C4×D3—68no—
81C3×Dic6—38no—
82C3×C3⋊C8—68no—
83#455—91noSylow
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—42no—
86C2×H(3,𝔽3)#12011yescharacteristic
87#272—42no—
88C3×#41—92no—
89C3×#41—182no—
90C2×C3×Q8—34no—
91C2×C3×Q8C2×C3⋊D311yescharacteristic
92C2×C3×D4—34no—
93C2×C3×D4—182no—
94C2×C3×D4—182no—
95C2×C3×D4C2×C3⋊D311yescharacteristic
96C2×#67—68no—
97#237—68no—
98D3×D4—68no—
99#235—38no—
100C3×Q16—182no—
101C3×SD16—182no—
102C3×M16—92no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4—68no—
105C3×C42—34no—
106C3×C42C2×C3⋊D311yescharacteristic
107#216—68no—
108#215—68no—
109#208—38no—
110C22×C32—28no—
111C22×C32—48no—
112C22×C32C2×#6714yes—
113C2×C3×D3—68no—
114C2×C3×D3—248no—
115C32×C4—44no—
116C32×C4—44no—
117C32×C4D3×D418yes—
118C3×Dic3—68no—
119H(5,𝔽2)—91no—
120#137—92no—
121#124#27211yescharacteristic, complemented
122#104—92no—
123#100—91no—
124H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
125C23×C3—38no—
126C23×C3—182no—
127C23×C3#63812yes—
128C22×D3—68no—
129C3×Q8—22no—
130C3×Q8—68no—
131C3×Q8—92no—
132C3×D4—22no—
133C3×D4—68no—
134C3×D4—92no—
135C3×D4—184no—
136C3×D4—182no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—34no—
140C2×C3×C4—64no—
141C2×C3×C4—64no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4C22×C3⋊D311yescharacteristic
145#67—68no—
146#67—616no—
147C2×Dic3—68no—
148D12—38no—
149C4×D3—68no—
150Dic6—38no—
151C3×C8—182no—
152C3⋊C8—68no—
153C2×C32—24no—
154C2×C32—48no—
155C2×C32#93714yes—
156C3×D3—128no—
157#41—92no—
158#41—182no—
159C2×Q8C2×#27211yescharacteristic, complemented
160C2×D4—182no—
161C2×D4—182no—
162C2×D4C2×#27211yescharacteristic
163Q16—182no—
164SD16—182no—
165M16—92no—
166C22⋊C4—22no—
167C42C2×#27211yescharacteristic, complemented
168C22×C3—22no—
169C22×C3—34no—
170C22×C3—42no—
171C22×C3—68no—
172C22×C3—128no—
173C22×C3—182no—
174C22×C3—722no—
175C22×C3C2×#63811yescharacteristic
176C2×D3—68no—
177C2×D3—248no—
178C3×C4—38no—
179C3×C4—41no—
180C3×C4—41no—
181C3×C4—124no—
182C3×C4—124no—
183C3×C4—182no—
184C3×C4D4×C3⋊D312yes—
185Dic3—68no—
186C32#506114yescomplemented
187C23—182no—
188C23#623512yes—
189Q8—22no—
190Q8—92no—
191D4—22no—
192D4—92no—
193D4—184no—
194D4—182no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—182no—
198C2×C4—182no—
199C2×C4C22×#27211yescharacteristic
200C8—182no—
201C2×C3—21no—
202C2×C3—34no—
203C2×C3—42no—
204C2×C3—64no—
205C2×C3—128no—
206C2×C3—362no—
207C2×C3#6414611yescharacteristic
208D3—128no—
209C22—22no—
210C22—42no—
211C22—182no—
212C22—722no—
213C22C2×#623511yescharacteristic
214C4—41no—
215C4—41no—
216C4—182no—
217C4D4×#27212yes—
218C3—34no—
219C3#1059276011yescharacteristic
220C2—21no—
221C2—42no—
222C2—362no—
223C2#1172184711yescharacteristic
subgroupG/H
center, socleC2×C3#64146
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#124C2
prime radical#11722664C2
hypercenterC3×#124C3⋊D3
hypocenterH(3,𝔽3)#455
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#124

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. H(3,𝔽3)×#124C2semidirect
3. #11721859C2semidirect
4. #11720579C2semidirect
5. #11720522C2semidirect
7. C2×Q8×H(3,𝔽3)C22semidirect
14. C42×H(3,𝔽3)C22semidirect
17. #88538C22semidirect
19. C32×#124D3semidirect
39. C2×C32×Q8C2×D3semidirect
46. C32×C42C2×D3semidirect
121. #124#272semidirect
124. H(3,𝔽3)#455semidirect
159. C2×Q8C2×#272semidirect
167. C42C2×#272semidirect
186. C32#5061semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117226642
H(3,𝔽3)×#1242
#117218592
#117205792
#117205222
C3×#50613